Improving efficiency of healthcare systems is a top national interest worldwide. However, the need of delivering scalable healthcare services to the patients while reducing costs is a challenging issue. Among the most promising approaches for enabling smart healthcare (s-health) are edge-computing capabilities and next-generation wireless networking technologies that can provide real-time and cost-effective patient remote monitoring. In this paper, we present our vision of exploiting multi-access edge computing (MEC) for s-health applications. We envision a MEC-based architecture and discuss the benefits that it can bring to realize in-network and context-aware processing so that the s-health requirements are met. We then present two main functionalities that can be implemented leveraging such an architecture to provide efficient data delivery, namely, multimodal data compression and edge-based feature extraction for event detection. The former allows efficient and low distortion compression, while the latter ensures high-reliability and fast response in case of emergency applications. Finally, we discuss the main challenges and opportunities that edge computing could provide and possible directions for future research.